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PROFESSOR: Today I want to
talk a little bit about
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designing control systems.
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This will finish up
our discussions
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on signals and systems.
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Let me then just briefly
review where we are.
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Hopefully this might help you
also for perspective with
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regard to thinking about
the exam tonight.
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We've looked at a bunch
of different kinds of
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representations for discrete
time systems.
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00:00:50,890 --> 00:00:55,320
The easiest, most concise method
we looked at was the
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representation using a
difference equation.
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00:00:58,700 --> 00:01:03,690
That's mathematically as
concise as you can get.
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00:01:03,690 --> 00:01:07,500
But it doesn't tell you
important things like who's
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the input and who's the output
and what are all the different
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ways that you can get
through the system
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00:01:12,620 --> 00:01:14,290
from input to output?
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00:01:14,290 --> 00:01:18,670
So for that question block
diagrams are nice.
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Block diagrams are graphical.
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It makes it very easy to see
when there is, for example, a
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00:01:24,500 --> 00:01:29,830
cyclic path through
the network.
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00:01:29,830 --> 00:01:30,780
But they're graphic.
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00:01:30,780 --> 00:01:34,100
They're not nearly so concise
as difference equations.
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00:01:34,100 --> 00:01:36,220
So then we went on
to operators.
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00:01:36,220 --> 00:01:42,780
Operators are just as concise
as difference equations but
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00:01:42,780 --> 00:01:46,400
they contain additional
information because the
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operators have an implicit
argument.
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00:01:49,890 --> 00:01:52,500
So there's an input, which is
the argument to the operator,
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00:01:52,500 --> 00:01:55,220
and there's an output, which is
the output of the operator.
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00:01:55,220 --> 00:01:59,210
So you can tell who is the input
and who is the output.
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00:01:59,210 --> 00:02:00,340
So that's good.
40
00:02:00,340 --> 00:02:03,040
That sort of combines the
strengths of difference
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00:02:03,040 --> 00:02:05,340
equations and block diagrams.
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00:02:05,340 --> 00:02:07,530
You end up with a concise
representation that has
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00:02:07,530 --> 00:02:12,660
complete information about
the signal flow paths.
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00:02:12,660 --> 00:02:17,700
Furthermore, you can analyze
the operators by using
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00:02:17,700 --> 00:02:23,210
polynomial mathematics and that
gives rise to the notion
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of a system functional.
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00:02:25,870 --> 00:02:29,470
And that's a very nice closure
because that represents an
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abstraction that lets us think
about a whole system as though
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00:02:34,590 --> 00:02:40,740
it were just one part, one
thing, one operator.
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00:02:40,740 --> 00:02:44,390
So we use that structure then,
all of those representations,
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00:02:44,390 --> 00:02:47,730
to try to learn about
feedback.
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00:02:47,730 --> 00:02:50,820
And first off, in the block
diagram it's very easy to see
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that any time you have feedback
-- feedback so
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enormously powerful that we want
to use it in design --
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but you can see immediately
from the structure of the
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block diagram that if you have
feedback then you have cycles.
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Why is that interesting?
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00:03:04,880 --> 00:03:08,060
Well, that's interesting because
if you have cycles
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then even transient inputs can
generate persistent outputs.
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So that's a kind of behavior
that we would like to
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understand.
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From the nature of feedback
it generates cycles.
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From the nature of cycles it
generates persistent responses
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even if there's no input.
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And we saw that we could
characterize those by thinking
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about those responses for
one part at a time.
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And those parts we thought of as
poles and the responses to
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a single pole we called modes.
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So we thought through a way of
decomposing the response of a
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complicated system in terms
of a number of additive
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components that are
based on poles.
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Poles are just the base of
a geometric sequence.
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So today then what I want to
do is use that framework to
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think about design.
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How would you optimize the
design of a controller?
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00:04:08,650 --> 00:04:11,210
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Looking back to where we've
been, way back in Lab 4,
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ancient history, we looked at
how you could program the
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robot to approach a wall.
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And we saw that depending on how
you set up that system you
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00:04:24,330 --> 00:04:27,920
could get very different
performances.
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And what we'd like to do is have
a way that we can design
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for performance without
actually building it.
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The kind of thing that we built
in the lab, Lab 4, was
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so easy that building it to
determine its behaviors was
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00:04:41,180 --> 00:04:42,290
not a bad problem.
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But In general if you were
building a 777, there's more
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00:04:47,960 --> 00:04:49,800
than one pole.
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00:04:49,800 --> 00:04:54,280
And you wouldn't necessarily
want to test drive all of the
90
00:04:54,280 --> 00:04:56,820
bad configurations.
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00:04:56,820 --> 00:04:59,070
So we'd like to be able to
understand that kind of a
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00:04:59,070 --> 00:05:01,200
problem analytically.
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00:05:01,200 --> 00:05:04,170
We'd like to be able
to analyze it.
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00:05:04,170 --> 00:05:08,790
So using the different
representations you can
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generate a very concise
representation just thinking
96
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in terms of difference
equations.
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00:05:12,580 --> 00:05:15,310
You all did this in Lab 4.
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00:05:15,310 --> 00:05:18,950
And you get a single difference
equation that tells
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00:05:18,950 --> 00:05:21,540
you in principle everything
there is that you could know,
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00:05:21,540 --> 00:05:23,650
but not in a form that's
very easy to analyze.
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00:05:23,650 --> 00:05:27,520
102
00:05:27,520 --> 00:05:29,400
It's a bit better if you
translate the difference
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00:05:29,400 --> 00:05:33,510
equation into a block diagram
because now you can see that
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00:05:33,510 --> 00:05:36,220
this system of equations
has in fact two
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00:05:36,220 --> 00:05:37,200
feedback loops in it.
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Two cycles.
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00:05:38,200 --> 00:05:41,570
Two things that might
potentially generate
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00:05:41,570 --> 00:05:44,820
persistent responses to
transient signals, which could
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00:05:44,820 --> 00:05:46,770
then degrade performance.
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00:05:46,770 --> 00:05:50,480
If the transient signal lasts
ten years it might be a bad
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00:05:50,480 --> 00:05:51,020
controller.
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00:05:51,020 --> 00:05:57,870
If the 777 hits turbulence
and never stabilizes
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00:05:57,870 --> 00:06:00,060
that would be bad.
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00:06:00,060 --> 00:06:04,950
If small disturbances got bigger
with time that might be
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00:06:04,950 --> 00:06:06,770
bad, right?
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00:06:06,770 --> 00:06:10,740
So we can see that this simple
controller described by these
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00:06:10,740 --> 00:06:13,110
difference equations has
the potential to do
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00:06:13,110 --> 00:06:14,180
that sort of stuff.
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00:06:14,180 --> 00:06:18,780
And we'd like to understand,
when does it?
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The easiest way to think about
analyzing this is to focus
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00:06:22,560 --> 00:06:27,810
first on the inner loop and ask
the question, what's the
122
00:06:27,810 --> 00:06:31,520
functional representation for
that box which we would call
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00:06:31,520 --> 00:06:32,660
an accumulator?
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00:06:32,660 --> 00:06:38,040
This box, this thing,
accumulates at its output, the
125
00:06:38,040 --> 00:06:41,110
sum of all the things that ever
came in so we call it an
126
00:06:41,110 --> 00:06:43,160
accumulator.
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00:06:43,160 --> 00:06:45,420
So what's the functional
representation of an
128
00:06:45,420 --> 00:06:46,440
accumulator?
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00:06:46,440 --> 00:06:48,160
Well, we just do polynomial
math.
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00:06:48,160 --> 00:06:54,120
Easy so we can recognize from
the block diagram that the
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00:06:54,120 --> 00:07:00,540
signal Y could be constructed
by applying R to W.
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00:07:00,540 --> 00:07:03,590
But we can also see in the block
diagram that W is the
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00:07:03,590 --> 00:07:09,680
sum of X and Y. And then if we
take the left hand side and
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00:07:09,680 --> 00:07:13,010
the right hand side of this
double equation we get
135
00:07:13,010 --> 00:07:16,100
something that involves just X
and Y, which we can solve for
136
00:07:16,100 --> 00:07:20,530
the ratio Y over X. Which then
says that the ratio is R over
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00:07:20,530 --> 00:07:22,550
(1 minus R).
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00:07:22,550 --> 00:07:25,025
That's a functional
representation for the effect
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00:07:25,025 --> 00:07:27,910
of the accumulation.
140
00:07:27,910 --> 00:07:31,460
That's also something that comes
up so frequently in the
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00:07:31,460 --> 00:07:35,450
design of control systems
that we give it a name.
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00:07:35,450 --> 00:07:38,550
We call this Black's Equation.
143
00:07:38,550 --> 00:07:43,120
And it's especially useful to
avoid these little trivial
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00:07:43,120 --> 00:07:46,930
steps in algebra, to just
jump to the answer.
145
00:07:46,930 --> 00:07:48,630
So let's see that everybody's
following me.
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00:07:48,630 --> 00:07:51,740
147
00:07:51,740 --> 00:07:54,070
The equation for this box is
the thing that we will call
148
00:07:54,070 --> 00:07:55,710
Black's Equation.
149
00:07:55,710 --> 00:07:56,980
It's not mysterious.
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00:07:56,980 --> 00:08:01,270
It's something that you could
derive, so derive it.
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00:08:01,270 --> 00:08:04,350
Figure out the functional form
for the system that goes from
152
00:08:04,350 --> 00:08:10,330
X to Y and figure out which
of these forms is correct.
153
00:08:10,330 --> 00:08:13,730
(1) through (4), or (5) if none
of the above applies.
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00:08:13,730 --> 00:08:16,120
So take 30 seconds, figure
out the answer.
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00:08:16,120 --> 00:08:17,390
I'm going to ask you
to raise your hand.
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00:08:17,390 --> 00:08:18,640
You're free to talk
to your neighbors.
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00:08:18,640 --> 00:09:11,990
158
00:09:11,990 --> 00:09:12,060
OK.
159
00:09:12,060 --> 00:09:13,560
So everybody who raised your
hands tell me what the
160
00:09:13,560 --> 00:09:15,610
right answer is.
161
00:09:15,610 --> 00:09:16,130
OK, wonderful.
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00:09:16,130 --> 00:09:17,360
It's about 95%.
163
00:09:17,360 --> 00:09:17,770
No.
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00:09:17,770 --> 00:09:20,463
It's about 100% correct out of
about 95% participation.
165
00:09:20,463 --> 00:09:22,990
166
00:09:22,990 --> 00:09:26,800
So the answer you can form just
like we did before, no
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00:09:26,800 --> 00:09:29,250
particular tricks, using
simple algebra.
168
00:09:29,250 --> 00:09:32,530
Simple algebra you get
F over (1 minus FG).
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00:09:32,530 --> 00:09:34,820
The thing I want you to
recognize is kind of a
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graphical way of thinking
about that.
171
00:09:37,610 --> 00:09:39,630
And you can just memorize
F over (1 minus
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00:09:39,630 --> 00:09:41,720
FG) and that's fine.
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00:09:41,720 --> 00:09:44,540
But there's some interesting
things that the designer
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thinks about.
175
00:09:47,020 --> 00:09:51,180
This functional form is F,
that's the forward gain.
176
00:09:51,180 --> 00:09:54,055
That's the gain through the
system starting at the input
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00:09:54,055 --> 00:09:57,230
and going directly
to the output.
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00:09:57,230 --> 00:10:04,930
So this form says that the
closed loop gain, the system
179
00:10:04,930 --> 00:10:11,350
function that results when the
loop is closed, is just the
180
00:10:11,350 --> 00:10:15,820
forward gain, F, divided by
(1 minus the loop gain).
181
00:10:15,820 --> 00:10:21,340
The loop gain is the product
around the loop once.
182
00:10:21,340 --> 00:10:23,050
So that's just a convenient
way of thinking about it.
183
00:10:23,050 --> 00:10:26,660
Any time you have a feedback
system of this type you can
184
00:10:26,660 --> 00:10:28,830
think about the closed loop
response as being the forward
185
00:10:28,830 --> 00:10:34,660
path F divided by (1 minus
loop gain FG).
186
00:10:34,660 --> 00:10:37,810
So the answer was 1.
187
00:10:37,810 --> 00:10:40,930
And generally we'll see two
different ways of representing
188
00:10:40,930 --> 00:10:42,370
the system.
189
00:10:42,370 --> 00:10:44,620
Sometimes we'll represent it
with a plus as we did with the
190
00:10:44,620 --> 00:10:46,120
accumulator.
191
00:10:46,120 --> 00:10:47,960
Many times we'll represent
it with a minus.
192
00:10:47,960 --> 00:10:50,080
That's the way we think about
control problems.
193
00:10:50,080 --> 00:10:51,890
We put in the minus because we'd
like to think about the
194
00:10:51,890 --> 00:10:55,140
controller as trying to make
something go to 0.
195
00:10:55,140 --> 00:10:57,140
So when you take the difference,
that gives us an
196
00:10:57,140 --> 00:10:58,730
error signal.
197
00:10:58,730 --> 00:11:00,720
And then we can think about the
controller being the thing
198
00:11:00,720 --> 00:11:03,120
that drives that error to 0.
199
00:11:03,120 --> 00:11:06,450
But however you think about it,
there's two forms that are
200
00:11:06,450 --> 00:11:07,980
very closely related.
201
00:11:07,980 --> 00:11:10,420
They really differ by just a
minus sign which you could
202
00:11:10,420 --> 00:11:13,500
think of as just multiplying
G. So it's sort of like the
203
00:11:13,500 --> 00:11:17,020
right hand side is just
a minus G plugged into
204
00:11:17,020 --> 00:11:19,360
the left hand side.
205
00:11:19,360 --> 00:11:21,970
So then the way you use this
idea, you think about the
206
00:11:21,970 --> 00:11:24,690
block diagram and you say, OK I
can replace this thing with
207
00:11:24,690 --> 00:11:26,360
an equivalent system
which is R over (1
208
00:11:26,360 --> 00:11:32,480
minus R) and then repeat.
209
00:11:32,480 --> 00:11:36,500
So the R over (1 minus R) means
that, if you think about
210
00:11:36,500 --> 00:11:39,890
Black's Equation now for this
loop, you should think about
211
00:11:39,890 --> 00:11:46,840
the forward gain, K minus T, R
over (1 minus R), that's this.
212
00:11:46,840 --> 00:11:49,240
Divided by (1 plus
the loop gain).
213
00:11:49,240 --> 00:11:53,820
So 1 plus, and the loop gain,
well, the wire down here just
214
00:11:53,820 --> 00:11:55,480
has a gain of 1.
215
00:11:55,480 --> 00:11:56,625
So the loop gain and
the forward gain
216
00:11:56,625 --> 00:11:58,500
are the same thing.
217
00:11:58,500 --> 00:12:01,170
So you get this kind
of expression which
218
00:12:01,170 --> 00:12:02,200
simplifies to this.
219
00:12:02,200 --> 00:12:04,890
There's two things I want
you to see about this.
220
00:12:04,890 --> 00:12:10,270
First off, I want you to see
that even though the
221
00:12:10,270 --> 00:12:13,460
simple-minded way of plugging
in said that we should have
222
00:12:13,460 --> 00:12:20,420
got a quotient of quotients N
over D over N over D, it's
223
00:12:20,420 --> 00:12:22,690
simplified to a single ratio.
224
00:12:22,690 --> 00:12:25,430
225
00:12:25,430 --> 00:12:29,700
If you design a system out of
just adders, gains and delays,
226
00:12:29,700 --> 00:12:31,550
that will always be true.
227
00:12:31,550 --> 00:12:33,740
There's a closure.
228
00:12:33,740 --> 00:12:37,750
It will always be the case
that the functional that
229
00:12:37,750 --> 00:12:40,560
represents a system of that
form will always have the
230
00:12:40,560 --> 00:12:43,920
property that it's a polynomial
in R divided by
231
00:12:43,920 --> 00:12:45,820
another polynomial in R,
that's just the way
232
00:12:45,820 --> 00:12:48,100
polynomials work.
233
00:12:48,100 --> 00:12:51,200
The other thing is that you
can now start to interpret
234
00:12:51,200 --> 00:12:56,140
what the behavior of this could
represent in a simpler
235
00:12:56,140 --> 00:12:59,740
form than thinking about this.
236
00:12:59,740 --> 00:13:03,820
So this kind of a representation
that leads to
237
00:13:03,820 --> 00:13:06,590
intuition about what the
behavior should be is very
238
00:13:06,590 --> 00:13:08,510
helpful when you're thinking
about design.
239
00:13:08,510 --> 00:13:13,850
And in particular this
particular thing says that if
240
00:13:13,850 --> 00:13:18,800
we think about a simpler system
that could generate
241
00:13:18,800 --> 00:13:23,690
that same response we can
generate some intuition for
242
00:13:23,690 --> 00:13:27,410
how we would like to set
the parameter k.
243
00:13:27,410 --> 00:13:33,030
So in particular this system
functional which we generated
244
00:13:33,030 --> 00:13:38,060
for this system could equally
apply to that system.
245
00:13:38,060 --> 00:13:41,200
246
00:13:41,200 --> 00:13:42,450
Is that clear?
247
00:13:42,450 --> 00:13:44,900
248
00:13:44,900 --> 00:13:50,610
So there's a numerator, which
I've represented here.
249
00:13:50,610 --> 00:13:52,810
There's a numerator, which
has an R in it and has
250
00:13:52,810 --> 00:13:54,010
a minus kT in it.
251
00:13:54,010 --> 00:13:56,890
I'm going to, for reasons that
you'll see in a minute, I'm
252
00:13:56,890 --> 00:13:59,360
going to call something P0
because it's the pole.
253
00:13:59,360 --> 00:14:02,710
254
00:14:02,710 --> 00:14:05,620
So the numerator I've
represented here and this
255
00:14:05,620 --> 00:14:09,160
denominator has this form.
256
00:14:09,160 --> 00:14:11,830
And I wrote it that way because
this is the canonical
257
00:14:11,830 --> 00:14:14,000
form for the way of thinking
about a pole.
258
00:14:14,000 --> 00:14:16,530
259
00:14:16,530 --> 00:14:19,120
So what I showed is that even
though this was a more
260
00:14:19,120 --> 00:14:24,910
complicated system you can think
about it as the cascade
261
00:14:24,910 --> 00:14:29,395
of a delay, a gain,
and a pole.
262
00:14:29,395 --> 00:14:33,010
263
00:14:33,010 --> 00:14:36,270
The pole can be calculated from
the gain, the pole is the
264
00:14:36,270 --> 00:14:41,890
multiplier for R, so the
pole is (1 plus kT).
265
00:14:41,890 --> 00:14:46,350
So if I were to choose kT to
be minus 0.2 then the pole
266
00:14:46,350 --> 00:14:49,460
would be at 0.8.
267
00:14:49,460 --> 00:14:55,170
If the pole is at 0.8 then the
mode, the natural response to
268
00:14:55,170 --> 00:14:59,070
the pole, would have the
form P to the n.
269
00:14:59,070 --> 00:15:02,590
They always have the form
geometric P to the n, so it
270
00:15:02,590 --> 00:15:05,490
would look like 0.8 to the n.
271
00:15:05,490 --> 00:15:09,030
Because of the pre-multiplier of
1 minus P0 the whole thing
272
00:15:09,030 --> 00:15:12,460
gets multiplied by 0.2 and
because of the delay the whole
273
00:15:12,460 --> 00:15:15,240
thing gets shifted
to the right.
274
00:15:15,240 --> 00:15:19,440
The important thing is that by
thinking about manipulating
275
00:15:19,440 --> 00:15:24,070
this as an operator we can
recognize and simplify the
276
00:15:24,070 --> 00:15:25,580
form of the behavior.
277
00:15:25,580 --> 00:15:27,650
That gives us an intuitive
grasp over how
278
00:15:27,650 --> 00:15:29,170
to best choose kT.
279
00:15:29,170 --> 00:15:31,710
280
00:15:31,710 --> 00:15:32,960
It's all clear?
281
00:15:32,960 --> 00:15:35,270
282
00:15:35,270 --> 00:15:38,940
Now, the behaviors that we're
interested in are not always
283
00:15:38,940 --> 00:15:40,160
unit-sample responses.
284
00:15:40,160 --> 00:15:42,500
We do unit-sample responses
because they're the easiest
285
00:15:42,500 --> 00:15:44,800
possible thing we could
think of, right?
286
00:15:44,800 --> 00:15:51,230
A unit-sample is the simplest
non-zero signal.
287
00:15:51,230 --> 00:15:54,980
A unit-sample is the signal that
is different from 0 in
288
00:15:54,980 --> 00:15:56,980
exactly one place --
289
00:15:56,980 --> 00:15:59,950
the easiest possible place, 0.
290
00:15:59,950 --> 00:16:01,600
And it has its easy-as-possible
291
00:16:01,600 --> 00:16:02,830
non-zero value --
292
00:16:02,830 --> 00:16:04,060
1.
293
00:16:04,060 --> 00:16:06,590
So we focus on the unit-sample
signal because it's the
294
00:16:06,590 --> 00:16:09,090
easiest possible signal
we could think about.
295
00:16:09,090 --> 00:16:10,960
But when we're thinking about
behaviors we're often thinking
296
00:16:10,960 --> 00:16:13,810
about other things.
297
00:16:13,810 --> 00:16:16,900
Often we'll think about
the step response.
298
00:16:16,900 --> 00:16:18,300
Here I have illustrated
the way you would
299
00:16:18,300 --> 00:16:20,790
measure a step response.
300
00:16:20,790 --> 00:16:28,240
A step response is what would
happen if the output were
301
00:16:28,240 --> 00:16:35,080
initially 0, if we were at rest,
and suddenly we turned
302
00:16:35,080 --> 00:16:39,620
on a signal that was
constant at 1.
303
00:16:39,620 --> 00:16:42,740
That would happen in the robot
case if we started the robot
304
00:16:42,740 --> 00:16:46,730
close to the wall, at
rest, near 0 --
305
00:16:46,730 --> 00:16:53,220
so that the output signal is
close to 0, but the desired
306
00:16:53,220 --> 00:16:54,680
input was a meter behind.
307
00:16:54,680 --> 00:16:57,550
308
00:16:57,550 --> 00:17:00,350
Then that would be an input
signal that started at time
309
00:17:00,350 --> 00:17:04,230
equals 0 equal to 1 and
persisted forever at 1.
310
00:17:04,230 --> 00:17:05,910
And the result then
would be what we
311
00:17:05,910 --> 00:17:08,230
call the step response.
312
00:17:08,230 --> 00:17:12,490
The step response is typically
easier to measure in the lab
313
00:17:12,490 --> 00:17:13,910
than is the unit-sample
response.
314
00:17:13,910 --> 00:17:15,880
So we use the unit-sample
response when we're thinking
315
00:17:15,880 --> 00:17:18,270
analytically, when we're doing
calculations, and we use the
316
00:17:18,270 --> 00:17:19,440
step response when
we're in the lab
317
00:17:19,440 --> 00:17:20,690
trying to measure something.
318
00:17:20,690 --> 00:17:23,760
319
00:17:23,760 --> 00:17:27,089
And the whole theory wouldn't
be very useful if there
320
00:17:27,089 --> 00:17:30,780
weren't a close relationship
between those two things.
321
00:17:30,780 --> 00:17:33,780
This diagram illustrates
the relationship.
322
00:17:33,780 --> 00:17:38,020
If we think about a system H
for which we would like to
323
00:17:38,020 --> 00:17:41,930
find the step response, the step
response to that system
324
00:17:41,930 --> 00:17:43,930
is what would the system
do if you put the
325
00:17:43,930 --> 00:17:47,080
unit-step into the system.
326
00:17:47,080 --> 00:17:49,290
I've represented the unit-step
here as u[n].
327
00:17:49,290 --> 00:17:52,220
328
00:17:52,220 --> 00:17:57,480
u[n], the signal that is 0 for
n less than 0, and 1 for n
329
00:17:57,480 --> 00:18:00,220
bigger than or equal to 0 --
330
00:18:00,220 --> 00:18:03,570
is just the accumulation of
the delta function, the
331
00:18:03,570 --> 00:18:06,290
unit-sample.
332
00:18:06,290 --> 00:18:13,420
So this system, the cascade of
an accumulator with H, would
333
00:18:13,420 --> 00:18:17,910
measure the step response of H
if it were driven with the
334
00:18:17,910 --> 00:18:19,160
sample signal.
335
00:18:19,160 --> 00:18:21,490
336
00:18:21,490 --> 00:18:26,480
Because of the properties of
polynomials and because block
337
00:18:26,480 --> 00:18:31,920
diagrams follow the rules for
polynomials, we can flip these
338
00:18:31,920 --> 00:18:37,750
whenever the systems both start
at rest, and if we flip
339
00:18:37,750 --> 00:18:41,630
those we get a different
interpretation.
340
00:18:41,630 --> 00:18:46,260
What this says is that if you
were to take H and stimulate
341
00:18:46,260 --> 00:18:50,750
it with the unit-sample you
would get h, little h, which
342
00:18:50,750 --> 00:18:53,000
we would call the unit-sample
response because it's the
343
00:18:53,000 --> 00:18:57,010
response to the system when the
input is the unit-sample.
344
00:18:57,010 --> 00:19:00,480
So if you measured h with the
unit-sample rather then with
345
00:19:00,480 --> 00:19:05,520
the unit-step you would get the
unit-sample response from
346
00:19:05,520 --> 00:19:08,590
which you could generate the
step response by running it
347
00:19:08,590 --> 00:19:09,840
through an accumulator.
348
00:19:09,840 --> 00:19:11,980
349
00:19:11,980 --> 00:19:15,040
So what that says is there's a
close association, there's a
350
00:19:15,040 --> 00:19:17,430
close relationship, between the
unit-sample response and
351
00:19:17,430 --> 00:19:18,780
the unit-step response.
352
00:19:18,780 --> 00:19:22,060
One is the accumulation
of the other.
353
00:19:22,060 --> 00:19:24,410
The unit-step response is the
accumulated response to the
354
00:19:24,410 --> 00:19:25,660
unit-sample response.
355
00:19:25,660 --> 00:19:28,020
356
00:19:28,020 --> 00:19:32,140
So that means that in that
previous example where we saw
357
00:19:32,140 --> 00:19:36,280
that setting kT equal to minus
0.2 resulted in this
358
00:19:36,280 --> 00:19:39,970
unit-sample response, that
would correspond to this
359
00:19:39,970 --> 00:19:41,400
unit-step response.
360
00:19:41,400 --> 00:19:46,630
All you do is for every sample
you calculate, for this
361
00:19:46,630 --> 00:19:52,140
response you calculate the sum
of say, n equals 0, you would
362
00:19:52,140 --> 00:19:57,360
take the sum of all of the
previous answers in H[n].
363
00:19:57,360 --> 00:19:59,520
It's the accumulation.
364
00:19:59,520 --> 00:20:04,830
So it starts at 0 since the sum
of all those numbers is 0.
365
00:20:04,830 --> 00:20:09,930
Then at time 1 it becomes the
sum from here back so it
366
00:20:09,930 --> 00:20:11,860
becomes 0.2.
367
00:20:11,860 --> 00:20:14,160
Then here it's the sum
from here back.
368
00:20:14,160 --> 00:20:16,750
And if you add these all up
it becomes a number that
369
00:20:16,750 --> 00:20:17,860
approaches 1.
370
00:20:17,860 --> 00:20:19,250
Not surprisingly, right?
371
00:20:19,250 --> 00:20:22,460
If you've got a feedback system
and if you started the
372
00:20:22,460 --> 00:20:25,710
robot up against the wall and
the desired position was one
373
00:20:25,710 --> 00:20:31,020
meter behind you would
monotonically approach 1, OK?
374
00:20:31,020 --> 00:20:33,350
And what you can see is that if
you change the value of the
375
00:20:33,350 --> 00:20:40,590
pole, here I've changed the kT
from minus 0.2 which is what
376
00:20:40,590 --> 00:20:44,360
the previous answer was, to
minus 0.8, I've changed the
377
00:20:44,360 --> 00:20:47,880
value of the pole,
the unit-sample
378
00:20:47,880 --> 00:20:49,300
response got faster.
379
00:20:49,300 --> 00:20:52,860
380
00:20:52,860 --> 00:20:55,290
And the unit-step response
also got faster.
381
00:20:55,290 --> 00:20:59,970
The point is there are different
kinds of performance
382
00:20:59,970 --> 00:21:03,300
metrics that we might want to
use, unit-sample response,
383
00:21:03,300 --> 00:21:07,260
unit-step response, but the
responses of all of them, you
384
00:21:07,260 --> 00:21:11,580
can tell something about the
response to all of them from
385
00:21:11,580 --> 00:21:13,350
the response of the unit-sample
signal.
386
00:21:13,350 --> 00:21:16,000
That's why we focus so much
on the unit-sample signal.
387
00:21:16,000 --> 00:21:17,740
It's not because it's the
most popular thing
388
00:21:17,740 --> 00:21:18,840
to use in the lab.
389
00:21:18,840 --> 00:21:22,830
It's because it's the easiest
thing to calculate with and it
390
00:21:22,830 --> 00:21:25,000
gives us insight into things
that we would like to measure
391
00:21:25,000 --> 00:21:27,690
in the lab.
392
00:21:27,690 --> 00:21:33,360
So for this very simple system
what you can show is that
393
00:21:33,360 --> 00:21:36,360
there's only a few possible
behaviors, a few
394
00:21:36,360 --> 00:21:40,160
categories of behaviors.
395
00:21:40,160 --> 00:21:46,370
If you were to choose kT to be
between 0 and 1, then the
396
00:21:46,370 --> 00:21:51,285
pole, which is (1 plus kT) would
also be between 0 and 1.
397
00:21:51,285 --> 00:21:54,150
398
00:21:54,150 --> 00:21:58,840
Since the system has a single
pole you can say a lot about
399
00:21:58,840 --> 00:22:02,930
the response from the numerical
value of the pole.
400
00:22:02,930 --> 00:22:05,540
If the pole is between 0 and 1
then the response is going to
401
00:22:05,540 --> 00:22:08,700
be monotonic and converging.
402
00:22:08,700 --> 00:22:11,510
403
00:22:11,510 --> 00:22:15,530
That results because the
unit-sample response was
404
00:22:15,530 --> 00:22:21,650
positive only and decayed
towards 0.
405
00:22:21,650 --> 00:22:32,380
Because it's positive only
it means monotonic --
406
00:22:32,380 --> 00:22:35,080
goes to 0 and makes
it converge.
407
00:22:35,080 --> 00:22:39,550
So you can infer properties
about the unit-step's response
408
00:22:39,550 --> 00:22:42,620
from properties of the pole
just like we could infer
409
00:22:42,620 --> 00:22:45,720
properties of the unit-sample
response.
410
00:22:45,720 --> 00:22:49,520
If you changed kT to be between
minus 2 and 1 you get
411
00:22:49,520 --> 00:22:52,010
a P0 that's between
minus 1 and 0.
412
00:22:52,010 --> 00:22:53,260
Again that's just
that equation.
413
00:22:53,260 --> 00:22:55,750
414
00:22:55,750 --> 00:23:00,260
That says that the response
will be alternating.
415
00:23:00,260 --> 00:23:03,160
So the sign of the unit-sample
response goes
416
00:23:03,160 --> 00:23:04,410
positive then negative.
417
00:23:04,410 --> 00:23:07,560
418
00:23:07,560 --> 00:23:10,650
It still converges in the sense
that the unit sample
419
00:23:10,650 --> 00:23:13,840
response approaches 0.
420
00:23:13,840 --> 00:23:17,200
And what that means for the unit
step-response is that the
421
00:23:17,200 --> 00:23:20,190
unit-step response will
converge toward 1.
422
00:23:20,190 --> 00:23:28,730
423
00:23:28,730 --> 00:23:30,820
The sign doesn't alternate
around 0, the sign
424
00:23:30,820 --> 00:23:32,070
alternates around 1.
425
00:23:32,070 --> 00:23:35,170
426
00:23:35,170 --> 00:23:37,370
So again, you can infer the
properties of the unit-step
427
00:23:37,370 --> 00:23:39,710
response from the properties of
the unit-sample response.
428
00:23:39,710 --> 00:23:44,480
And if you have kT that is less
than minus 2 then you get
429
00:23:44,480 --> 00:23:46,300
a P0 that's less than
minus 1 and
430
00:23:46,300 --> 00:23:47,980
that's a divergent response.
431
00:23:47,980 --> 00:23:51,200
So the point is you can infer
properties about the control
432
00:23:51,200 --> 00:23:56,120
system by thinking about the
poles of the system where here
433
00:23:56,120 --> 00:23:57,950
I've illustrated it for
a simple system that
434
00:23:57,950 --> 00:23:59,200
only has one pole.
435
00:23:59,200 --> 00:24:01,780
436
00:24:01,780 --> 00:24:02,720
OK.
437
00:24:02,720 --> 00:24:03,760
I told you a bunch of facts.
438
00:24:03,760 --> 00:24:05,260
Now you figure out something.
439
00:24:05,260 --> 00:24:10,340
How would I choose k for this
system to get the quote, "best
440
00:24:10,340 --> 00:27:00,640
performance?"
441
00:27:00,640 --> 00:27:04,340
So which value of kT would give
the fastest convergence
442
00:27:04,340 --> 00:27:06,575
for the unit-sample signal?
443
00:27:06,575 --> 00:27:11,670
444
00:27:11,670 --> 00:27:18,980
OK, participation is down but
the hit rate is still good.
445
00:27:18,980 --> 00:27:20,980
Virtually everybody who
volunteered to answer got the
446
00:27:20,980 --> 00:27:23,580
right answer.
447
00:27:23,580 --> 00:27:25,055
The most popular
answer was (2).
448
00:27:25,055 --> 00:27:27,950
Why is the answer (2)?
449
00:27:27,950 --> 00:27:29,870
What's the range of
possibilities
450
00:27:29,870 --> 00:27:31,120
that we could get?
451
00:27:31,120 --> 00:27:35,560
452
00:27:35,560 --> 00:27:40,710
If we choose k, or kT, we
could choose kT to be--
453
00:27:40,710 --> 00:27:44,190
what's the range of kT
that we could use?
454
00:27:44,190 --> 00:27:46,260
Minus 2 to 0?
455
00:27:46,260 --> 00:27:51,060
[INAUDIBLE] we could use kT,
any real number, right?
456
00:27:51,060 --> 00:27:53,350
We couldn't use imaginary
numbers because that doesn't
457
00:27:53,350 --> 00:27:55,830
sort of make sense for
a real system.
458
00:27:55,830 --> 00:27:59,330
But we could choose
any real number.
459
00:27:59,330 --> 00:28:04,400
The real numbers map, according
to this chart, the
460
00:28:04,400 --> 00:28:06,710
real numbers map to a different
real number.
461
00:28:06,710 --> 00:28:09,670
If you choose kT you can figure
out where is the pole
462
00:28:09,670 --> 00:28:11,770
by that mapping.
463
00:28:11,770 --> 00:28:14,735
Where would you put the pole to
get the fastest response?
464
00:28:14,735 --> 00:28:19,660
465
00:28:19,660 --> 00:28:22,230
If you have your choice of
putting the pole anywhere on
466
00:28:22,230 --> 00:28:24,400
this red line, that red line or
that red line, where would
467
00:28:24,400 --> 00:28:28,970
you put it and why?
468
00:28:28,970 --> 00:28:33,870
469
00:28:33,870 --> 00:28:37,140
AUDIENCE: Just inside
the [INAUDIBLE]
470
00:28:37,140 --> 00:28:39,020
PROFESSOR: Putting it inside the
unit circle would probably
471
00:28:39,020 --> 00:28:40,862
be a good idea because?
472
00:28:40,862 --> 00:28:42,130
AUDIENCE: [UNINTELLIGIBLE]
473
00:28:42,130 --> 00:28:42,920
PROFESSOR: Yeah.
474
00:28:42,920 --> 00:28:44,170
If you didn't put it
inside the unit
475
00:28:44,170 --> 00:28:45,854
circle it wouldn't converge.
476
00:28:45,854 --> 00:28:48,060
That's right.
477
00:28:48,060 --> 00:28:49,820
So you like [UNINTELLIGIBLE]
the inside.
478
00:28:49,820 --> 00:28:52,450
Given the choice of anywhere
here and anywhere here which
479
00:28:52,450 --> 00:28:53,350
would you choose?
480
00:28:53,350 --> 00:28:54,050
How would you choose it?
481
00:28:54,050 --> 00:28:54,400
Yeah.
482
00:28:54,400 --> 00:28:55,280
AUDIENCE: Derive.
483
00:28:55,280 --> 00:28:57,414
PROFESSOR: Derive.
484
00:28:57,414 --> 00:28:58,908
And how do you get that?
485
00:28:58,908 --> 00:29:00,158
AUDIENCE: [UNINTELLIGIBLE]
486
00:29:00,158 --> 00:29:03,390
487
00:29:03,390 --> 00:29:04,884
PROFESSOR: What would happen
if it was close to
488
00:29:04,884 --> 00:29:06,134
[UNINTELLIGIBLE]?
489
00:29:06,134 --> 00:29:08,106
490
00:29:08,106 --> 00:29:10,630
It converges quite
quickly, right?
491
00:29:10,630 --> 00:29:13,470
Poles always converge
geometrically.
492
00:29:13,470 --> 00:29:16,600
The base of the geometric
is the pole value.
493
00:29:16,600 --> 00:29:19,150
So you'd like the pole to be
as small as possible to get
494
00:29:19,150 --> 00:29:22,140
the convergence as
fast as possible.
495
00:29:22,140 --> 00:29:24,620
That make sense to everybody?
496
00:29:24,620 --> 00:29:29,790
So in particular for this
example if you chose kT to be
497
00:29:29,790 --> 00:29:37,010
minus 1 in that limit then this
entire factor goes away.
498
00:29:37,010 --> 00:29:42,820
So the entire response
degenerates to R and R is not
499
00:29:42,820 --> 00:29:45,970
instantaneous but it's
pretty fast.
500
00:29:45,970 --> 00:29:49,730
What that says is that
you get to the final
501
00:29:49,730 --> 00:29:54,710
value in one step.
502
00:29:54,710 --> 00:29:58,860
So if the input consisted of
a unit sample, which has
503
00:29:58,860 --> 00:30:02,010
non-zero value only at 0, the
output would have non-zero
504
00:30:02,010 --> 00:30:09,150
value only at 1, right?
505
00:30:09,150 --> 00:30:13,440
Thinking about the way that
works in practice, think about
506
00:30:13,440 --> 00:30:15,310
the robot and think about
we're trying to
507
00:30:15,310 --> 00:30:17,000
drive toward the wall.
508
00:30:17,000 --> 00:30:23,570
If we made kT be minus one, and
just for the sake of being
509
00:30:23,570 --> 00:30:27,420
concrete let me say that
T is about 1/10.
510
00:30:27,420 --> 00:30:30,960
That's what the sampling period
is for the robots we
511
00:30:30,960 --> 00:30:32,880
use in the lab.
512
00:30:32,880 --> 00:30:38,720
If T were 1/10 then the best
k would be minus 10.
513
00:30:38,720 --> 00:30:42,550
And what that says is that if
we were 1 meter away from
514
00:30:42,550 --> 00:30:47,350
where we want to be we would set
the velocity to 10 meters
515
00:30:47,350 --> 00:30:48,620
per second.
516
00:30:48,620 --> 00:30:53,400
What that says is that if we
started 1 meter away from
517
00:30:53,400 --> 00:30:56,880
where we want to be, so this
is intended to represent
518
00:30:56,880 --> 00:31:01,390
position on the same axis that
this has showed, so if we
519
00:31:01,390 --> 00:31:07,630
started here and we wanted to be
here, time is plotted down.
520
00:31:07,630 --> 00:31:11,930
If we use the rule that we just
specified then we would
521
00:31:11,930 --> 00:31:15,400
set the velocity given this
condition which is 1 meter
522
00:31:15,400 --> 00:31:16,780
away from where we want to be.
523
00:31:16,780 --> 00:31:20,070
We would set the velocity
to be 10.
524
00:31:20,070 --> 00:31:25,380
If we set the velocity to be 10
then after 1 unit of time,
525
00:31:25,380 --> 00:31:28,820
after 1/10 of a second, we are
1 meter to the right, which
526
00:31:28,820 --> 00:31:31,130
just happens to be exactly
where we want to be.
527
00:31:31,130 --> 00:31:33,910
528
00:31:33,910 --> 00:31:36,920
Had we chosen k to be bigger
we would have overshot.
529
00:31:36,920 --> 00:31:39,950
Had we chosen k to be smaller
we would have undershot.
530
00:31:39,950 --> 00:31:45,170
k equals 10 gave us precisely
the right answer so that we
531
00:31:45,170 --> 00:31:47,020
get there in one fell swoop.
532
00:31:47,020 --> 00:31:50,240
Then on the very next step we
would compute a velocity of 0
533
00:31:50,240 --> 00:31:53,670
because we are at where
we want to be so
534
00:31:53,670 --> 00:31:56,080
we would stay there.
535
00:31:56,080 --> 00:31:59,240
And that condition would
persist forever.
536
00:31:59,240 --> 00:32:03,350
The idea would be this simple
system provides a way that we
537
00:32:03,350 --> 00:32:05,700
could set the gain so we could
get to where we want
538
00:32:05,700 --> 00:32:07,580
to be in one step.
539
00:32:07,580 --> 00:32:08,830
It's hard to beat that.
540
00:32:08,830 --> 00:32:12,010
541
00:32:12,010 --> 00:32:14,330
The problem that results and the
reason you didn't see that
542
00:32:14,330 --> 00:32:18,060
good behavior in the lab was
that the sensors in the robot
543
00:32:18,060 --> 00:32:21,260
don't work instantaneously.
544
00:32:21,260 --> 00:32:23,780
They introduce delay.
545
00:32:23,780 --> 00:32:26,920
And as an idealization of that
delay I want to think through
546
00:32:26,920 --> 00:32:28,250
the same problem.
547
00:32:28,250 --> 00:32:32,760
But now let's say that the
sensor delays the input to the
548
00:32:32,760 --> 00:32:36,100
sensor which is the output
of the system.
549
00:32:36,100 --> 00:32:40,980
Let's say that the sensor
introduces a delay of 1, so
550
00:32:40,980 --> 00:32:45,610
now instead of reporting d
sensed, which was d0[n], it
551
00:32:45,610 --> 00:32:46,860
reports d0[n minus 1].
552
00:32:46,860 --> 00:32:50,960
553
00:32:50,960 --> 00:32:54,090
So now what would happen?
554
00:32:54,090 --> 00:33:00,250
Now with the delay, if I started
here and if by some
555
00:33:00,250 --> 00:33:06,750
mysterious process I was here
at time n equals 1, then I
556
00:33:06,750 --> 00:33:10,080
would calculate my
new velocity.
557
00:33:10,080 --> 00:33:12,130
What would be my new
velocity here?
558
00:33:12,130 --> 00:33:14,590
I'm right where I want to be.
559
00:33:14,590 --> 00:33:17,564
What would be my new velocity
if I assume that the sensor
560
00:33:17,564 --> 00:33:21,460
has a delay of [UNINTELLIGIBLE]?
561
00:33:21,460 --> 00:33:21,947
AUDIENCE: 10.
562
00:33:21,947 --> 00:33:23,410
PROFESSOR: 10.
563
00:33:23,410 --> 00:33:27,920
Because of the delay the sensor
is reporting that I'm a
564
00:33:27,920 --> 00:33:31,850
meter away from where
I want to be.
565
00:33:31,850 --> 00:33:34,870
So the controller calculates,
oh, I need to
566
00:33:34,870 --> 00:33:35,990
go forward a meter.
567
00:33:35,990 --> 00:33:38,710
I'll set the velocity to 10.
568
00:33:38,710 --> 00:33:42,320
So having set the velocity to 10
and then one step goes by,
569
00:33:42,320 --> 00:33:46,330
now we're completely
on the wrong side.
570
00:33:46,330 --> 00:33:49,210
That's what happens when you
put delay into the system.
571
00:33:49,210 --> 00:33:52,970
So because we're basing this
decision on where we were last
572
00:33:52,970 --> 00:33:57,220
time we go to the wrong place.
573
00:33:57,220 --> 00:33:59,880
So now we're here.
574
00:33:59,880 --> 00:34:02,264
What will the controller
say next?
575
00:34:02,264 --> 00:34:03,150
AUDIENCE: [UNINTELLIGIBLE]
576
00:34:03,150 --> 00:34:04,480
PROFESSOR: Stay here.
577
00:34:04,480 --> 00:34:07,650
You're in a great place.
578
00:34:07,650 --> 00:34:09,810
I'm really 1 meter too close.
579
00:34:09,810 --> 00:34:12,429
In fact I banged
into the wall.
580
00:34:12,429 --> 00:34:15,469
But the sensor is telling me I'm
exactly where I want to be
581
00:34:15,469 --> 00:34:16,719
so stay here.
582
00:34:16,719 --> 00:34:20,080
583
00:34:20,080 --> 00:34:26,090
Say I didn't kill myself, what
will the velocity next be?
584
00:34:26,090 --> 00:34:27,199
Minus 10.
585
00:34:27,199 --> 00:34:29,610
So now I tell myself
to go back.
586
00:34:29,610 --> 00:34:30,860
That's probably a good move.
587
00:34:30,860 --> 00:34:33,620
588
00:34:33,620 --> 00:34:38,130
But now I still think I'm too
close to the wall so I tell
589
00:34:38,130 --> 00:34:40,889
myself to continue to back up.
590
00:34:40,889 --> 00:34:44,639
The idea is that I get
poor performance.
591
00:34:44,639 --> 00:34:48,920
The delay had a devastating
effect on the way that the
592
00:34:48,920 --> 00:34:50,790
controller worked.
593
00:34:50,790 --> 00:34:54,190
Even though it's a tiny change
to the way the system works it
594
00:34:54,190 --> 00:34:57,060
has a devastating effect
on behavior.
595
00:34:57,060 --> 00:34:59,400
We'd like to be able to predict
that without having to
596
00:34:59,400 --> 00:35:01,240
measure it.
597
00:35:01,240 --> 00:35:03,690
Here's the same equations except
that I put a delay in
598
00:35:03,690 --> 00:35:05,440
the sensor.
599
00:35:05,440 --> 00:35:08,300
Here is the same block diagram
but I've represented a delay
600
00:35:08,300 --> 00:35:09,550
in the sensor path.
601
00:35:09,550 --> 00:35:13,660
602
00:35:13,660 --> 00:35:17,060
So now the question is, what's
the new functional
603
00:35:17,060 --> 00:35:20,910
representation for that
control system?
604
00:35:20,910 --> 00:36:02,046
605
00:36:02,046 --> 00:37:53,740
606
00:37:53,740 --> 00:37:55,530
So what's the answer?
607
00:37:55,530 --> 00:37:59,365
Can you rate the functional form
for this system as one of
608
00:37:59,365 --> 00:38:02,195
(1), (2), (3), or (4), or
is it none of the above?
609
00:38:02,195 --> 00:38:07,200
610
00:38:07,200 --> 00:38:12,610
About 1/3 participation and
about 100% correct.
611
00:38:12,610 --> 00:38:13,860
The answer's four.
612
00:38:13,860 --> 00:38:16,360
613
00:38:16,360 --> 00:38:18,872
You get to use Black's Equation
or however you'd like
614
00:38:18,872 --> 00:38:20,530
to think about that.
615
00:38:20,530 --> 00:38:23,280
You can think about reducing the
inner loop the same as we
616
00:38:23,280 --> 00:38:26,930
did before and then think about
this as forward over (1
617
00:38:26,930 --> 00:38:30,760
plus loop gain) but now the loop
gain has R squared in it
618
00:38:30,760 --> 00:38:35,810
instead of R. We
get this form.
619
00:38:35,810 --> 00:38:39,410
How does this form differ from
when the R wasn't here?
620
00:38:39,410 --> 00:38:42,140
What's the difference between
R not there and R is there?
621
00:38:42,140 --> 00:38:49,020
622
00:38:49,020 --> 00:38:49,275
What's the answer?
623
00:38:49,275 --> 00:38:50,525
Anyone?
624
00:38:50,525 --> 00:38:54,415
625
00:38:54,415 --> 00:38:55,665
AUDIENCE: [UNINTELLIGIBLE]
626
00:38:55,665 --> 00:38:59,265
627
00:38:59,265 --> 00:39:00,235
PROFESSOR: The squared term.
628
00:39:00,235 --> 00:39:04,940
So this term in the previous
form was just an R and in this
629
00:39:04,940 --> 00:39:08,720
form is a square and
so what's that do?
630
00:39:08,720 --> 00:39:10,330
What's the importance
of the fact that
631
00:39:10,330 --> 00:39:13,660
there's a square there?
632
00:39:13,660 --> 00:39:15,790
Two poles, right?
633
00:39:15,790 --> 00:39:18,850
We now have a polynomial in
the denominator that is
634
00:39:18,850 --> 00:39:21,750
quadratic in R.
635
00:39:21,750 --> 00:39:23,560
And what that's going to do is
it's going to give us two
636
00:39:23,560 --> 00:39:26,650
poles instead of one.
637
00:39:26,650 --> 00:39:28,250
The importance of that is that
now we're going to have to
638
00:39:28,250 --> 00:39:31,140
think through-- we previously
categorized what were all the
639
00:39:31,140 --> 00:39:34,600
behaviors you could
get from one pole.
640
00:39:34,600 --> 00:39:39,050
The behaviors you can get from
one pole were monotonic
641
00:39:39,050 --> 00:39:45,590
divergence, non-monotonic
alternating divergence,
642
00:39:45,590 --> 00:39:51,840
monotonic convergence,
alternating convergence.
643
00:39:51,840 --> 00:39:53,220
So there were four behaviors
that were
644
00:39:53,220 --> 00:39:54,590
possible with one pole.
645
00:39:54,590 --> 00:39:56,390
Now we have to think through
what are all the possible
646
00:39:56,390 --> 00:39:58,780
behaviors that we could
get with two poles.
647
00:39:58,780 --> 00:40:00,750
Different problem.
648
00:40:00,750 --> 00:40:03,530
Hopefully they're related.
649
00:40:03,530 --> 00:40:08,480
So here, the way we would find
out what the poles are is take
650
00:40:08,480 --> 00:40:15,730
this expression, substitute for
every R, 1 over z, turn
651
00:40:15,730 --> 00:40:18,410
the ratio of polynomials
in R into a ratio of
652
00:40:18,410 --> 00:40:19,660
polynomials in z.
653
00:40:19,660 --> 00:40:22,700
654
00:40:22,700 --> 00:40:25,580
To do that in this case I had
to multiply numerator and
655
00:40:25,580 --> 00:40:26,880
denominator by z squared.
656
00:40:26,880 --> 00:40:29,830
657
00:40:29,830 --> 00:40:33,190
Having done that I get a second
order polynomial in z
658
00:40:33,190 --> 00:40:35,960
in the bottom so there's two
poles which are the roots of
659
00:40:35,960 --> 00:40:38,960
that polynomial.
660
00:40:38,960 --> 00:40:40,420
And that's just a quadratic
equation.
661
00:40:40,420 --> 00:40:43,100
662
00:40:43,100 --> 00:40:46,050
The interesting thing now is
to map out what are all the
663
00:40:46,050 --> 00:40:48,410
possible behaviors that that
system can give us.
664
00:40:48,410 --> 00:40:51,350
665
00:40:51,350 --> 00:40:54,470
It's important to realize that's
a simple generalization
666
00:40:54,470 --> 00:40:57,600
of what we saw before.
667
00:40:57,600 --> 00:41:01,090
It will be the case that any
system that we construct out
668
00:41:01,090 --> 00:41:06,330
of adders, gains and delays will
have the property that we
669
00:41:06,330 --> 00:41:09,010
can write the system functional
as a ratio of
670
00:41:09,010 --> 00:41:16,930
polynomials in R. By the factor
theorem we will always
671
00:41:16,930 --> 00:41:20,340
be able to factor
the denominator.
672
00:41:20,340 --> 00:41:23,270
And by the notion of partial
fractions we'll always be able
673
00:41:23,270 --> 00:41:26,660
to write some complicated
expression like that in terms
674
00:41:26,660 --> 00:41:28,960
of a sum of parts.
675
00:41:28,960 --> 00:41:31,700
Each part being first order.
676
00:41:31,700 --> 00:41:34,960
The intuition we get from this
is that what we ought to do is
677
00:41:34,960 --> 00:41:40,630
factor the denominator, find
the poles, and associate a
678
00:41:40,630 --> 00:41:44,090
behavior with each
of those poles.
679
00:41:44,090 --> 00:41:48,080
Here's what the problem looks
like for the two pole problem.
680
00:41:48,080 --> 00:41:52,450
If we have the general form
given here and if we start by
681
00:41:52,450 --> 00:41:59,760
thinking about kT having a small
magnitude, if kT has a
682
00:41:59,760 --> 00:42:02,300
small magnitude then we have 1/2
plus or minus the square
683
00:42:02,300 --> 00:42:05,150
root of 1/2 squared.
684
00:42:05,150 --> 00:42:06,680
So that's 1/2 plus
or minus 1/2.
685
00:42:06,680 --> 00:42:08,990
That's 0 or 1.
686
00:42:08,990 --> 00:42:13,470
So the poles for this system,
if you make k be very small,
687
00:42:13,470 --> 00:42:17,040
the poles are at 0,
near 0 and near 1.
688
00:42:17,040 --> 00:42:21,140
Is that a good system response
or a bad system response?
689
00:42:21,140 --> 00:42:21,550
Bad.
690
00:42:21,550 --> 00:42:22,800
Why?
691
00:42:22,800 --> 00:42:25,221
692
00:42:25,221 --> 00:42:27,797
Well, we're trying to think
through the behavior of the
693
00:42:27,797 --> 00:42:30,520
second order system by thinking
about the separate
694
00:42:30,520 --> 00:42:34,280
behaviors of each
of the poles.
695
00:42:34,280 --> 00:42:37,360
Is this a good pole
or a bad pole?
696
00:42:37,360 --> 00:42:39,860
Why?
697
00:42:39,860 --> 00:42:43,760
The response is always pole
[UNINTELLIGIBLE].
698
00:42:43,760 --> 00:42:48,800
The mode associated with the
response at a pole near one is
699
00:42:48,800 --> 00:42:51,380
something near one to the end.
700
00:42:51,380 --> 00:42:54,130
That never converges.
701
00:42:54,130 --> 00:42:57,520
If you start with some error
the error persists forever.
702
00:42:57,520 --> 00:43:00,030
Well, that's not good.
703
00:43:00,030 --> 00:43:06,410
If wind turbulence knocks you
into a decline in your
704
00:43:06,410 --> 00:43:12,230
airplane and it persists
forever, that's not good.
705
00:43:12,230 --> 00:43:14,440
You would like those
things to damp out.
706
00:43:14,440 --> 00:43:15,700
So this pole is bad.
707
00:43:15,700 --> 00:43:16,950
How about that pole?
708
00:43:16,950 --> 00:43:20,150
709
00:43:20,150 --> 00:43:22,560
That one has a response
that decays quickly.
710
00:43:22,560 --> 00:43:24,870
But the problem is that when
you add the two pieces
711
00:43:24,870 --> 00:43:26,660
together, that was the reason
I showed you this
712
00:43:26,660 --> 00:43:30,450
decomposition, you can think
about the polynomial being
713
00:43:30,450 --> 00:43:34,280
factored and being broken
into a number of parts.
714
00:43:34,280 --> 00:43:37,180
The part that's associated with
the pole near one has a
715
00:43:37,180 --> 00:43:39,590
response that goes
for a long time.
716
00:43:39,590 --> 00:43:44,390
So that will asymptotically
dominate your response.
717
00:43:44,390 --> 00:43:47,950
So we refer to this as
a dominant pole.
718
00:43:47,950 --> 00:43:51,070
This pole dominates
the response.
719
00:43:51,070 --> 00:43:55,040
That's a way of inferring the
behavior of two poles from the
720
00:43:55,040 --> 00:43:59,820
sum of single poles.
721
00:43:59,820 --> 00:44:01,970
In this particular case there's
one pole that matters
722
00:44:01,970 --> 00:44:03,170
more than the other one.
723
00:44:03,170 --> 00:44:06,280
So we call that pole
the dominant pole.
724
00:44:06,280 --> 00:44:11,990
If you were to make kT more
negative, So here's the
725
00:44:11,990 --> 00:44:12,720
general form.
726
00:44:12,720 --> 00:44:16,510
If you make kT negative, you can
make the thing under the
727
00:44:16,510 --> 00:44:20,240
radical sign go towards 0.
728
00:44:20,240 --> 00:44:24,400
If you made the thing under the
radical sign go to 0 then
729
00:44:24,400 --> 00:44:26,790
you would get two
poles at 1/2.
730
00:44:26,790 --> 00:44:30,780
731
00:44:30,780 --> 00:44:34,370
So here we would see that if kT
were minus 1/4, if kT were
732
00:44:34,370 --> 00:44:37,550
minus 1/4 we would have 1/2
squared, which is plus 1/4.
733
00:44:37,550 --> 00:44:41,150
Minus 1/4 would give us
0 under the radical.
734
00:44:41,150 --> 00:44:43,470
So we would get two
poles at 1/2.
735
00:44:43,470 --> 00:44:44,720
Is that good or bad?
736
00:44:44,720 --> 00:44:47,870
737
00:44:47,870 --> 00:44:51,510
Well, it's better than the
previous example, right?
738
00:44:51,510 --> 00:44:55,040
Because each of those poles is
associated with the response
739
00:44:55,040 --> 00:44:59,410
where the error gets half what
it used to be on every step.
740
00:44:59,410 --> 00:45:00,660
So it converges.
741
00:45:00,660 --> 00:45:06,560
742
00:45:06,560 --> 00:45:10,720
If going from 0 to minus 1/4 is
good, then going to minus
743
00:45:10,720 --> 00:45:12,900
1/2 might be better, right?
744
00:45:12,900 --> 00:45:19,680
If you continue that trend, say
you make kT be minus 1, if
745
00:45:19,680 --> 00:45:23,660
kT is minus 1 then you get
1/2 squared minus one.
746
00:45:23,660 --> 00:45:28,360
So 1/2 squared is 1/4, minus
1 would be minus 3/4.
747
00:45:28,360 --> 00:45:30,560
That gives us a complex
pole here.
748
00:45:30,560 --> 00:45:33,140
749
00:45:33,140 --> 00:45:38,280
So we get two poles that are
right on the unit circle.
750
00:45:38,280 --> 00:45:39,110
What's that mean?
751
00:45:39,110 --> 00:45:41,900
That means oscillations.
752
00:45:41,900 --> 00:45:46,070
Oscillations is something you
can't get with one pole with a
753
00:45:46,070 --> 00:45:48,240
real system.
754
00:45:48,240 --> 00:45:51,750
Oscillations result from
a poll that has
755
00:45:51,750 --> 00:45:53,360
an imaginary component.
756
00:45:53,360 --> 00:45:55,160
If the system is real
you could only get
757
00:45:55,160 --> 00:46:00,330
such poles in pairs.
758
00:46:00,330 --> 00:46:02,540
So it's this pair that
makes sense for
759
00:46:02,540 --> 00:46:05,750
a real valued system.
760
00:46:05,750 --> 00:46:07,520
And that gives rise to
oscillations and that's
761
00:46:07,520 --> 00:46:11,140
exactly what we saw here.
762
00:46:11,140 --> 00:46:15,030
So we can associate the
oscillations that we saw in
763
00:46:15,030 --> 00:46:21,090
the simulated lab experiment
with poles that have imaginary
764
00:46:21,090 --> 00:46:22,340
components.
765
00:46:22,340 --> 00:46:25,230
766
00:46:25,230 --> 00:46:28,550
So what would be the period of
the oscillation in the system
767
00:46:28,550 --> 00:46:33,620
given by 1/2 plus
j root 3 over 2?
768
00:46:33,620 --> 00:46:50,735
769
00:46:50,735 --> 00:46:52,202
AUDIENCE: [INAUDIBLE]
770
00:46:52,202 --> 00:46:54,464
PROFESSOR: Excuse me?
771
00:46:54,464 --> 00:46:55,398
Excuse me?
772
00:46:55,398 --> 00:46:57,740
AUDIENCE: Where did the
root three come from?
773
00:46:57,740 --> 00:47:01,010
PROFESSOR: The previous page.
774
00:47:01,010 --> 00:47:04,190
If you substitute minus 1.
775
00:47:04,190 --> 00:47:48,370
776
00:47:48,370 --> 00:47:49,625
So what's the period
of the oscillation?
777
00:47:49,625 --> 00:47:53,286
778
00:47:53,286 --> 00:47:57,171
So the period's represented
by 5 converged into 6.
779
00:47:57,171 --> 00:48:00,440
So how do you get 6?
780
00:48:00,440 --> 00:48:03,130
The easiest way to think about
that is to think about the
781
00:48:03,130 --> 00:48:07,020
poles being expressed
in polar notation.
782
00:48:07,020 --> 00:48:09,770
The poles we previously said
were 1/2 plus or minus
783
00:48:09,770 --> 00:48:11,330
j root 3 over 2.
784
00:48:11,330 --> 00:48:15,390
That's the same as e plus
or minus j pi over 3.
785
00:48:15,390 --> 00:48:20,300
It's easier to use that form
because if you take that form,
786
00:48:20,300 --> 00:48:25,020
so if you think about e to the
j, what was it, 2 pi over 3?
787
00:48:25,020 --> 00:48:27,760
788
00:48:27,760 --> 00:48:29,010
Pi over 3.
789
00:48:29,010 --> 00:48:33,240
790
00:48:33,240 --> 00:48:39,150
So if you think about that form,
that's the pole, we can
791
00:48:39,150 --> 00:48:42,550
write that that way.
792
00:48:42,550 --> 00:48:46,140
793
00:48:46,140 --> 00:48:50,190
Then the inside has
a magnitude of 1.
794
00:48:50,190 --> 00:48:53,360
So we can think about that just
being a magnitude of 1
795
00:48:53,360 --> 00:48:55,850
and an angle of pi over 3.
796
00:48:55,850 --> 00:49:00,010
So when you raise that to the n,
the magnitude to the n, one
797
00:49:00,010 --> 00:49:04,820
to the n is always 1, and the
angle raised to the n, it just
798
00:49:04,820 --> 00:49:07,010
increases linearly with n.
799
00:49:07,010 --> 00:49:12,230
So the angle goes from pi over
3 to 2pi over 3 to pi to 4pi
800
00:49:12,230 --> 00:49:15,030
over 3, et cetera.
801
00:49:15,030 --> 00:49:18,510
So you can think about this
going from pi over 3, 2pi over
802
00:49:18,510 --> 00:49:20,350
3, pi, 4, 5, 6.
803
00:49:20,350 --> 00:49:22,930
804
00:49:22,930 --> 00:49:25,990
It takes n equals 6 to get
around to where it started so
805
00:49:25,990 --> 00:49:27,240
the period is 6.
806
00:49:27,240 --> 00:49:32,750
807
00:49:32,750 --> 00:49:36,000
If you were to further change
the game, if you were to make
808
00:49:36,000 --> 00:49:40,960
it even more negative, the poles
would go outside the
809
00:49:40,960 --> 00:49:41,560
unit circle.
810
00:49:41,560 --> 00:49:44,100
And then what would happen?
811
00:49:44,100 --> 00:49:46,060
AUDIENCE: [UNINTELLIGIBLE]
812
00:49:46,060 --> 00:49:46,550
PROFESSOR: Right.
813
00:49:46,550 --> 00:49:49,500
So that's completely
unacceptable.
814
00:49:49,500 --> 00:49:52,470
The point is that by changing
the gain you can get any
815
00:49:52,470 --> 00:49:57,050
behavior on this figure which
is called the root locus.
816
00:49:57,050 --> 00:50:00,670
So root meaning the root
of a polynomial.
817
00:50:00,670 --> 00:50:04,480
Locus meaning the acceptable
values of points.
818
00:50:04,480 --> 00:50:08,280
So the root locus shows you
all the possible behaviors
819
00:50:08,280 --> 00:50:10,470
they could result from
this system.
820
00:50:10,470 --> 00:50:15,460
So given that root locus, how
would you choose k to make
821
00:50:15,460 --> 00:50:17,545
your system response as
fast as you could?
822
00:50:17,545 --> 00:51:19,830
823
00:51:19,830 --> 00:51:22,250
So what value of kT
would you want?
824
00:51:22,250 --> 00:51:23,500
Everyone raise your hands.
825
00:51:23,500 --> 00:51:26,122
826
00:51:26,122 --> 00:51:27,670
That's very good.
827
00:51:27,670 --> 00:51:31,780
So the most popular answer
is number (2).
828
00:51:31,780 --> 00:51:34,350
So why would the answer
be number (2)?
829
00:51:34,350 --> 00:51:35,600
What do you look at?
830
00:51:35,600 --> 00:51:39,500
831
00:51:39,500 --> 00:51:40,496
Yeah?
832
00:51:40,496 --> 00:51:43,484
AUDIENCE: [INAUDIBLE]
833
00:51:43,484 --> 00:51:44,980
PROFESSOR: Remember what
we're trying to do.
834
00:51:44,980 --> 00:51:48,500
We're trying to infer properties
of the behavior of
835
00:51:48,500 --> 00:51:53,530
this second order system from
the pole locations.
836
00:51:53,530 --> 00:51:57,120
We know that there's an
expansion that lets us expand
837
00:51:57,120 --> 00:52:01,930
the system in terms of the sum
of two first order responses.
838
00:52:01,930 --> 00:52:06,130
The slowest of the first order
responses will dominate
839
00:52:06,130 --> 00:52:07,860
eventually.
840
00:52:07,860 --> 00:52:09,720
So what we need to
look at is the
841
00:52:09,720 --> 00:52:13,570
slowest of the two responses.
842
00:52:13,570 --> 00:52:15,940
We would like to know
of the two poles
843
00:52:15,940 --> 00:52:18,400
which one is the slowest?
844
00:52:18,400 --> 00:52:23,220
The slowest is the one that's
closest to the unit circle.
845
00:52:23,220 --> 00:52:27,690
So we would get the fastest
response when the slowest one
846
00:52:27,690 --> 00:52:29,110
is as fast as possible.
847
00:52:29,110 --> 00:52:32,040
848
00:52:32,040 --> 00:52:39,480
As the poles initially go toward
each other from 0 to 1
849
00:52:39,480 --> 00:52:42,470
this one is getting faster, this
one is getting slower.
850
00:52:42,470 --> 00:52:45,460
So the slowest one
is this one.
851
00:52:45,460 --> 00:52:49,590
So the slowest one is fastest
when they meet.
852
00:52:49,590 --> 00:52:52,320
And then when they diverge
does the slowest one get
853
00:52:52,320 --> 00:52:54,636
faster or slower?
854
00:52:54,636 --> 00:52:56,500
It's already slower
because it gets
855
00:52:56,500 --> 00:52:59,980
closer to the unit circle.
856
00:52:59,980 --> 00:53:02,610
So you get the fastest response
whenever you get the
857
00:53:02,610 --> 00:53:08,050
two poles both colliding at 1/2
and that was the case that
858
00:53:08,050 --> 00:53:13,610
happened when kT was minus 1/4
from two or three slides ago.
859
00:53:13,610 --> 00:53:18,910
So the idea then is to try to
infer what would be the
860
00:53:18,910 --> 00:53:22,430
behavior of this higher order
system by thinking about the
861
00:53:22,430 --> 00:53:27,190
behaviors of the individual
components, here the poles.
862
00:53:27,190 --> 00:53:29,960
And what we saw was something
that's in fact a very
863
00:53:29,960 --> 00:53:31,500
important general trend.
864
00:53:31,500 --> 00:53:35,600
What we saw was that we first
analyzed the wall finder
865
00:53:35,600 --> 00:53:39,110
system assuming there was
no delay in the sensor.
866
00:53:39,110 --> 00:53:42,130
And we found that that system
was characterized by a single
867
00:53:42,130 --> 00:53:45,870
pole and we had the design
freedom of putting that pole
868
00:53:45,870 --> 00:53:49,650
anywhere we wanted to
on the real axis.
869
00:53:49,650 --> 00:53:54,390
And that allowed us to choose
the pole to be at 0 which gave
870
00:53:54,390 --> 00:53:57,200
terrific performance.
871
00:53:57,200 --> 00:53:59,580
The interesting thing that
happened when you add just one
872
00:53:59,580 --> 00:54:02,910
more pole by putting a delay
in the sensor, you make the
873
00:54:02,910 --> 00:54:05,560
system more complicated and
now you can't possibly get
874
00:54:05,560 --> 00:54:08,030
nearly so good behavior.
875
00:54:08,030 --> 00:54:11,460
The behavior is a lot worse
than it was before.
876
00:54:11,460 --> 00:54:15,390
And in fact, if you were to do
the same kind of analysis by
877
00:54:15,390 --> 00:54:22,550
putting yet another delay in the
sensor you would find even
878
00:54:22,550 --> 00:54:24,550
worse behavior.
879
00:54:24,550 --> 00:54:27,810
The idea then, the
generalization of the way the
880
00:54:27,810 --> 00:54:31,070
behaviors is working, generally
speaking adding
881
00:54:31,070 --> 00:54:35,650
delays inside a feedback loop
is a destabilizing thing.
882
00:54:35,650 --> 00:54:40,440
Generally as the number of
delays increases you end up
883
00:54:40,440 --> 00:54:44,730
having to back off on the
maximum gain that you can use
884
00:54:44,730 --> 00:54:49,020
because the system becomes
less stable.
885
00:54:49,020 --> 00:54:54,440
So the overall moral is that
delays are bad generally.
886
00:54:54,440 --> 00:54:56,560
I mean, you could concoct some
kind of a weird scheme where
887
00:54:56,560 --> 00:54:57,460
that wouldn't be true.
888
00:54:57,460 --> 00:54:59,820
But it's actually hard to
concoct such a weird scheme.
889
00:54:59,820 --> 00:55:02,270
In general, and in virtually
every physical system that
890
00:55:02,270 --> 00:55:05,090
you'll run into, adding
delays makes the
891
00:55:05,090 --> 00:55:07,030
system harder to stabilize.
892
00:55:07,030 --> 00:55:09,010
And that's the big message.
893
00:55:09,010 --> 00:55:13,390
And the system that we looked at
in the lab, the wall finder
894
00:55:13,390 --> 00:55:17,190
was actually quite hard
because the number
895
00:55:17,190 --> 00:55:18,820
of delays was large.
896
00:55:18,820 --> 00:55:21,810
If you try to track where delays
can enter the robot
897
00:55:21,810 --> 00:55:25,460
system they get in at very
many different places.
898
00:55:25,460 --> 00:55:28,270
In the physical sensor, in the
microprocessor, in the
899
00:55:28,270 --> 00:55:31,230
conversion from analog to
digital, there's a number of
900
00:55:31,230 --> 00:55:32,850
delays in that system.
901
00:55:32,850 --> 00:55:36,200
And that's why it becomes
hard to stabilize.
902
00:55:36,200 --> 00:55:36,780
OK.
903
00:55:36,780 --> 00:55:39,450
So that's the main content
for today.
904
00:55:39,450 --> 00:55:42,150
What I want to do is give you
one more practice question.
905
00:55:42,150 --> 00:55:45,040
906
00:55:45,040 --> 00:55:47,710
The big problem that I want you
to think about from today
907
00:55:47,710 --> 00:55:51,870
is how do you characterize
performance?
908
00:55:51,870 --> 00:55:55,940
When we had a single pole
performance was easy to talk
909
00:55:55,940 --> 00:56:02,170
about because performance was
diverging monotonically,
910
00:56:02,170 --> 00:56:07,730
diverging alternating,
converging monotonically,
911
00:56:07,730 --> 00:56:09,220
converging alternating.
912
00:56:09,220 --> 00:56:10,970
There were four kinds
of behaviors.
913
00:56:10,970 --> 00:56:13,750
When we went to second order
we saw some new behaviors.
914
00:56:13,750 --> 00:56:17,230
It could become oscillatory.
915
00:56:17,230 --> 00:56:20,040
What I'd like you to do now is
think not just about those
916
00:56:20,040 --> 00:56:21,600
properties but many
other properties.
917
00:56:21,600 --> 00:56:23,190
So here's some questions.
918
00:56:23,190 --> 00:56:26,150
Think about the system on the
top and I'd like you to infer
919
00:56:26,150 --> 00:56:28,950
properties about that system.
920
00:56:28,950 --> 00:56:34,350
In particular, does this system
have three poles?
921
00:56:34,350 --> 00:56:37,360
Is the unit sample response, is
there a way to write that
922
00:56:37,360 --> 00:56:41,630
as the sum of three geometric
sequences?
923
00:56:41,630 --> 00:56:44,430
What's the unit sample
response?
924
00:56:44,430 --> 00:56:47,220
And is one of the poles
that z equals 1?
925
00:56:47,220 --> 00:56:50,080
So think about the system,
think about five ways of
926
00:56:50,080 --> 00:56:52,890
characterizing it and tell
me how many of those five
927
00:56:52,890 --> 00:56:54,230
characterizations is correct.
928
00:56:54,230 --> 01:00:58,700
929
01:00:58,700 --> 01:01:00,045
So how many of the properties
are true?
930
01:01:00,045 --> 01:01:03,733
931
01:01:03,733 --> 01:01:04,983
AUDIENCE: [UNINTELLIGIBLE]
932
01:01:04,983 --> 01:01:12,535
933
01:01:12,535 --> 01:01:15,480
Probably 2/3 correct?
934
01:01:15,480 --> 01:01:16,730
PROFESSOR: How many poles?
935
01:01:16,730 --> 01:01:20,190
936
01:01:20,190 --> 01:01:23,355
How do you get three?
937
01:01:23,355 --> 01:01:24,834
Where are the poles?
938
01:01:24,834 --> 01:01:27,792
AUDIENCE: [UNINTELLIGIBLE]
939
01:01:27,792 --> 01:01:28,285
PROFESSOR: How do
I find poles?
940
01:01:28,285 --> 01:01:30,257
What do I do?
941
01:01:30,257 --> 01:01:31,736
Yes?
942
01:01:31,736 --> 01:01:35,187
AUDIENCE: You use Black's
Equation [UNINTELLIGIBLE]
943
01:01:35,187 --> 01:01:38,110
the top you can express
[UNINTELLIGIBLE]
944
01:01:38,110 --> 01:01:41,540
so use Black's Equation to
express the system function as
945
01:01:41,540 --> 01:01:44,480
R cubed over (1 minus
R-cubed) then--
946
01:01:44,480 --> 01:01:44,920
PROFESSOR: That's right.
947
01:01:44,920 --> 01:01:49,680
AUDIENCE: --the denominator
as an order of 3 and
948
01:01:49,680 --> 01:01:52,080
you combine the 3s.
949
01:01:52,080 --> 01:01:55,060
PROFESSOR: So a little more
formally, we would take this
950
01:01:55,060 --> 01:02:01,160
thing and we would rewrite that
with R goes to 1 over z.
951
01:02:01,160 --> 01:02:09,070
So we get 1 over z cubed, 1
minus (1 over z cubed), which
952
01:02:09,070 --> 01:02:12,810
is then, clearing the z cubes
we would get 1 over
953
01:02:12,810 --> 01:02:14,820
(z cubed minus 1).
954
01:02:14,820 --> 01:02:17,340
955
01:02:17,340 --> 01:02:19,290
How many poles?
956
01:02:19,290 --> 01:02:19,890
Three.
957
01:02:19,890 --> 01:02:22,774
What are the poles of
z cubed minus 1?
958
01:02:22,774 --> 01:02:25,750
959
01:02:25,750 --> 01:02:28,230
Three poles in what?
960
01:02:28,230 --> 01:02:29,718
AUDIENCE: 0 [UNINTELLIGIBLE]
961
01:02:29,718 --> 01:02:33,190
962
01:02:33,190 --> 01:02:33,686
PROFESSOR: And so let's vote.
963
01:02:33,686 --> 01:02:35,174
Let's take a vote.
964
01:02:35,174 --> 01:02:39,142
There's two poles
at z equals 1.
965
01:02:39,142 --> 01:02:41,622
Yes?
966
01:02:41,622 --> 01:02:42,614
No?
967
01:02:42,614 --> 01:02:43,606
AUDIENCE: [UNINTELLIGIBLE]
968
01:02:43,606 --> 01:02:45,094
PROFESSOR: Why not?
969
01:02:45,094 --> 01:02:46,344
AUDIENCE: [UNINTELLIGIBLE]
970
01:02:46,344 --> 01:02:48,100
971
01:02:48,100 --> 01:02:51,050
PROFESSOR: So there's
two poles.
972
01:02:51,050 --> 01:02:53,550
I made a [UNINTELLIGIBLE]
plane.
973
01:02:53,550 --> 01:02:54,800
Where's the poles?
974
01:02:54,800 --> 01:02:57,340
975
01:02:57,340 --> 01:03:00,140
Well, you could factor
it, right?
976
01:03:00,140 --> 01:03:02,690
If you factored it you'd
find that there is
977
01:03:02,690 --> 01:03:06,680
a pole at 1, right?
978
01:03:06,680 --> 01:03:10,170
But then there's two more
poles like that.
979
01:03:10,170 --> 01:03:14,480
980
01:03:14,480 --> 01:03:20,290
So the poles are the three
roots of 1, which can be
981
01:03:20,290 --> 01:03:30,210
written like 1 e to the j, 2pi
over 3, and e to the j minus
982
01:03:30,210 --> 01:03:31,460
2pi over 3.
983
01:03:31,460 --> 01:03:35,520
984
01:03:35,520 --> 01:03:37,070
Which pole was the
dominant pole?
985
01:03:37,070 --> 01:03:38,320
AUDIENCE: [UNINTELLIGIBLE]
986
01:03:38,320 --> 01:03:43,022
987
01:03:43,022 --> 01:03:46,130
PROFESSOR: OK, bad question.
988
01:03:46,130 --> 01:03:49,016
What's a better question?
989
01:03:49,016 --> 01:03:50,350
AUDIENCE: How many dominant
poles are there?
990
01:03:50,350 --> 01:03:51,970
PROFESSOR: How many dominant
poles are there?
991
01:03:51,970 --> 01:03:53,100
That's a much better
question, yes.
992
01:03:53,100 --> 01:03:55,130
There's sort of three
poles that are
993
01:03:55,130 --> 01:03:56,630
equally dominant, right?
994
01:03:56,630 --> 01:04:00,420
They all have the
same magnitude.
995
01:04:00,420 --> 01:04:02,000
Why do we talk about
dominant poles?
996
01:04:02,000 --> 01:04:03,250
What are dominant
poles good for?
997
01:04:03,250 --> 01:04:07,610
998
01:04:07,610 --> 01:04:12,510
If I told you that I had a pole
at 3 and a pole at minus
999
01:04:12,510 --> 01:04:14,110
1, which one's the
dominant pole?
1000
01:04:14,110 --> 01:04:14,590
AUDIENCE: 3.
1001
01:04:14,590 --> 01:04:16,510
PROFESSOR: Why?
1002
01:04:16,510 --> 01:04:17,000
AUDIENCE: Greater magnitude.
1003
01:04:17,000 --> 01:04:18,620
PROFESSOR: Greater magnitude.
1004
01:04:18,620 --> 01:04:21,440
Why do we care?
1005
01:04:21,440 --> 01:04:24,300
We don't care, right?
1006
01:04:24,300 --> 01:04:25,930
What's good about the
dominant pole?
1007
01:04:25,930 --> 01:04:31,690
1008
01:04:31,690 --> 01:04:35,320
Well, we can write this response
as something that
1009
01:04:35,320 --> 01:04:39,400
looks like three to the n
plus minus 1 to the n.
1010
01:04:39,400 --> 01:04:42,260
If you let n get big enough
the only one that
1011
01:04:42,260 --> 01:04:45,240
matters is 3 to the n.
1012
01:04:45,240 --> 01:04:49,210
So if all you care about is
exactly how the plane was
1013
01:04:49,210 --> 01:04:53,710
flying the instant before it hit
the ground then you would
1014
01:04:53,710 --> 01:04:55,990
only need to worry
about long time.
1015
01:04:55,990 --> 01:04:58,150
And if you only worry about
long time you only need to
1016
01:04:58,150 --> 01:05:02,030
worry about the pole that's
worst behaved.
1017
01:05:02,030 --> 01:05:03,770
That's where the concept
comes from.
1018
01:05:03,770 --> 01:05:06,740
So none of these poles are
particularly worse behaved
1019
01:05:06,740 --> 01:05:08,810
than the others.
1020
01:05:08,810 --> 01:05:10,510
What's the unit-sample response
1021
01:05:10,510 --> 01:05:11,760
associated with that pole?
1022
01:05:11,760 --> 01:05:17,420
1023
01:05:17,420 --> 01:05:20,998
We have a name for that
[INAUDIBLE] right?
1024
01:05:20,998 --> 01:05:21,992
AUDIENCE: It's huge.
1025
01:05:21,992 --> 01:05:25,040
PROFESSOR: It's
[UNINTELLIGIBLE].
1026
01:05:25,040 --> 01:05:26,520
What's the unit-sample response
1027
01:05:26,520 --> 01:05:27,780
associated with this pole?
1028
01:05:27,780 --> 01:05:33,280
1029
01:05:33,280 --> 01:05:37,466
Well, it's got a complex
value right?
1030
01:05:37,466 --> 01:05:40,870
So the unit-sample response
associated with that pole is e
1031
01:05:40,870 --> 01:05:47,050
to the j 2 pi over 3 n.
1032
01:05:47,050 --> 01:05:49,430
That's a complex number.
1033
01:05:49,430 --> 01:05:56,040
That's 1 at time 0 and e to the
j 2 pi over 3n at time 1
1034
01:05:56,040 --> 01:06:01,420
and e to the j 4 pi over
3 at time two.
1035
01:06:01,420 --> 01:06:04,950
So it goes from here at 0 to
here at 1 to here at 2,3, 4,
1036
01:06:04,950 --> 01:06:06,200
5, 6, 7, 8.
1037
01:06:06,200 --> 01:06:08,950
1038
01:06:08,950 --> 01:06:15,100
What's the period
of this pole?
1039
01:06:15,100 --> 01:06:18,230
What's the period of the
unit-sample responses
1040
01:06:18,230 --> 01:06:20,550
associated with that pole?
1041
01:06:20,550 --> 01:06:21,440
3.
1042
01:06:21,440 --> 01:06:23,580
Because it takes 3
to get around to
1043
01:06:23,580 --> 01:06:24,650
where you started again.
1044
01:06:24,650 --> 01:06:28,392
What's the period
of this pole?
1045
01:06:28,392 --> 01:06:28,853
3.
1046
01:06:28,853 --> 01:06:30,697
You just spin around backward.
1047
01:06:30,697 --> 01:06:31,485
What's the period
of the response
1048
01:06:31,485 --> 01:06:34,746
associated with that pole?
1049
01:06:34,746 --> 01:06:36,162
Bad question.
1050
01:06:36,162 --> 01:06:38,530
All right, what's a
better question?
1051
01:06:38,530 --> 01:06:40,254
Is there a period associated
with it?
1052
01:06:40,254 --> 01:06:42,614
You could say that period
1 [UNINTELLIGIBLE]
1053
01:06:42,614 --> 01:06:43,864
definition of period.
1054
01:06:43,864 --> 01:06:46,390
1055
01:06:46,390 --> 01:06:48,350
What's the period
of this pole?
1056
01:06:48,350 --> 01:06:51,822
1057
01:06:51,822 --> 01:06:55,130
Dumb question, right?
1058
01:06:55,130 --> 01:06:58,150
Period implies repeat.
1059
01:06:58,150 --> 01:07:01,840
If the response repeats itself
after some time then we would
1060
01:07:01,840 --> 01:07:03,090
say the response is periodic.
1061
01:07:03,090 --> 01:07:06,450
1062
01:07:06,450 --> 01:07:09,270
Neither of those poles,
well, the minus 1 is.
1063
01:07:09,270 --> 01:07:13,020
Is the minus 1 pole periodic?
1064
01:07:13,020 --> 01:07:15,285
Yes.
1065
01:07:15,285 --> 01:07:18,907
What's the difference between
periodic and alternation?
1066
01:07:18,907 --> 01:07:21,160
Does a [UNINTELLIGIBLE]
1067
01:07:21,160 --> 01:07:22,090
alternate?
1068
01:07:22,090 --> 01:07:23,070
AUDIENCE: Yes.
1069
01:07:23,070 --> 01:07:24,215
PROFESSOR: Does it oscillate?
1070
01:07:24,215 --> 01:07:26,019
AUDIENCE: No.
1071
01:07:26,019 --> 01:07:28,420
PROFESSOR: Bad question.
1072
01:07:28,420 --> 01:07:33,410
Alternate is a word that we
invented for one pole.
1073
01:07:33,410 --> 01:07:35,730
Because the response
alternated in sine.
1074
01:07:35,730 --> 01:07:39,350
The unit-sample response in one
pole, where the pole is a
1075
01:07:39,350 --> 01:07:41,670
negative number, alternated
in sine.
1076
01:07:41,670 --> 01:07:44,790
So we gave that a name.
1077
01:07:44,790 --> 01:07:47,310
Alternation is not necessarily
something that we would like
1078
01:07:47,310 --> 01:07:50,210
to associate with a higher
order system.
1079
01:07:50,210 --> 01:07:53,220
Periodic is perfectly reasonable
to talk about for a
1080
01:07:53,220 --> 01:07:54,580
higher order system.
1081
01:07:54,580 --> 01:07:58,920
Periodic merely means that if y
of n were a periodic signal
1082
01:07:58,920 --> 01:08:04,360
that I could express y of
n plus n as y of n.
1083
01:08:04,360 --> 01:08:05,610
That would be periodic.
1084
01:08:05,610 --> 01:08:07,610
1085
01:08:07,610 --> 01:08:10,320
If the thing repeats itself we
would say it's periodic.
1086
01:08:10,320 --> 01:08:12,650
So the point of going over this
stuff is just to give you
1087
01:08:12,650 --> 01:08:15,670
some exercise in thinking
about how to think about
1088
01:08:15,670 --> 01:08:18,109
properties of systems.
1089
01:08:18,109 --> 01:08:19,830
We develop properties initially
1090
01:08:19,830 --> 01:08:23,290
thinking about one pole.
1091
01:08:23,290 --> 01:08:26,010
Those properties were easy.
1092
01:08:26,010 --> 01:08:30,910
Converging, diverging,
monotonic, alternating.
1093
01:08:30,910 --> 01:08:34,729
When we try to think about
corresponding properties of
1094
01:08:34,729 --> 01:08:38,040
higher order systems we can't
simply map the simple
1095
01:08:38,040 --> 01:08:40,090
properties of first order
systems into the other.
1096
01:08:40,090 --> 01:08:42,189
We have to think about more
complicated things.
1097
01:08:42,189 --> 01:08:44,520
Then we think about things
like dominant.
1098
01:08:44,520 --> 01:08:46,760
If one of the poles has a bigger
magnitude than the
1099
01:08:46,760 --> 01:08:49,870
other then for large times we
can ignore the smaller one.
1100
01:08:49,870 --> 01:08:55,420
1101
01:08:55,420 --> 01:08:56,795
What happens for short time?
1102
01:08:56,795 --> 01:09:00,149
1103
01:09:00,149 --> 01:09:03,850
Does this response monotonically
increase with
1104
01:09:03,850 --> 01:09:05,210
time for all time?
1105
01:09:05,210 --> 01:09:11,792
1106
01:09:11,792 --> 01:09:13,250
No.
1107
01:09:13,250 --> 01:09:16,892
Since the response associated
with minus 1 alternates in
1108
01:09:16,892 --> 01:09:23,399
sine, for short times, for times
with n close to 0, that
1109
01:09:23,399 --> 01:09:26,750
can be just as important
as this one.
1110
01:09:26,750 --> 01:09:32,439
So the dominant pole idea tells
you how things work when
1111
01:09:32,439 --> 01:09:34,590
you have large times.
1112
01:09:34,590 --> 01:09:36,819
It doesn't necessarily tell you
how things work when you
1113
01:09:36,819 --> 01:09:38,020
have small times.
1114
01:09:38,020 --> 01:09:42,060
How about, the unit sample
response is the sum of three
1115
01:09:42,060 --> 01:09:42,790
geometrics.
1116
01:09:42,790 --> 01:09:44,040
Yes or no?
1117
01:09:44,040 --> 01:09:46,865
1118
01:09:46,865 --> 01:09:48,320
What are the three geometrics?
1119
01:09:48,320 --> 01:09:53,170
1120
01:09:53,170 --> 01:09:54,625
And the answer to that's yes.
1121
01:09:54,625 --> 01:09:55,110
That's very important.
1122
01:09:55,110 --> 01:10:01,720
The three geometrics over here
are this pole to the n plus
1123
01:10:01,720 --> 01:10:08,830
this pole to the n plus
something that goes with this
1124
01:10:08,830 --> 01:10:12,770
other pole to the n.
1125
01:10:12,770 --> 01:10:14,840
Now it's a weighted sum but
the weights are not
1126
01:10:14,840 --> 01:10:16,090
necessarily 0.
1127
01:10:16,090 --> 01:10:21,120
1128
01:10:21,120 --> 01:10:23,440
The slide that I showed you
for the partial fraction
1129
01:10:23,440 --> 01:10:26,540
decomposition, you can always
write a higher order system as
1130
01:10:26,540 --> 01:10:29,460
a sum of first order factors.
1131
01:10:29,460 --> 01:10:31,940
That's the partial fraction
expansion.
1132
01:10:31,940 --> 01:10:34,070
That doesn't mean the weights
are all unity.
1133
01:10:34,070 --> 01:10:36,830
1134
01:10:36,830 --> 01:10:39,720
Number (2), can you write the
unit-sample response as the
1135
01:10:39,720 --> 01:10:41,470
sum of three geometric
signals?
1136
01:10:41,470 --> 01:10:41,860
Yes.
1137
01:10:41,860 --> 01:10:44,290
There it is.
1138
01:10:44,290 --> 01:10:46,390
And if you're really good at
complex math you could find
1139
01:10:46,390 --> 01:10:48,530
out a, b and c.
1140
01:10:48,530 --> 01:10:50,960
And that would tell you the
unit-sample response and that
1141
01:10:50,960 --> 01:10:52,570
would tell you the answer
to (3) and (4).
1142
01:10:52,570 --> 01:10:54,407
Is the unit-sample response 0,
0, 0, 1, 0, 0, 0, 1, 0, 0, 0,
1143
01:10:54,407 --> 01:10:56,738
1, 0, 0, 0, 1?
1144
01:10:56,738 --> 01:10:59,410
Or 1, 0, 0, 0, 1, 1 --
1145
01:10:59,410 --> 01:11:01,110
whatever.
1146
01:11:01,110 --> 01:11:05,140
Is it one of those two or
something different?
1147
01:11:05,140 --> 01:11:09,076
And how do you figure it out?
1148
01:11:09,076 --> 01:11:10,060
How do you figure
out the unit?
1149
01:11:10,060 --> 01:11:11,310
Is the unit-sample response--
1150
01:11:11,310 --> 01:11:13,560
1151
01:11:13,560 --> 01:11:14,524
Is number (2) correct?
1152
01:11:14,524 --> 01:11:17,174
Is the unit-sample response
0, 0, 0, 1, 0, 0,
1153
01:11:17,174 --> 01:11:18,424
0, 1, 0, 0, 0, 1--
1154
01:11:18,424 --> 01:11:21,680
1155
01:11:21,680 --> 01:11:23,390
Yes?
1156
01:11:23,390 --> 01:11:26,100
How do you know that?
1157
01:11:26,100 --> 01:11:27,841
You could solve that equation.
1158
01:11:27,841 --> 01:11:32,611
1159
01:11:32,611 --> 01:11:34,719
Is there an easier way?
1160
01:11:34,719 --> 01:11:36,156
Yeah?
1161
01:11:36,156 --> 01:11:38,551
AUDIENCE: I wrote it as
a difference equation.
1162
01:11:38,551 --> 01:11:39,988
PROFESSOR: Just write the
difference equation.
1163
01:11:39,988 --> 01:11:40,946
Exactly.
1164
01:11:40,946 --> 01:11:43,820
So even though I bad-mouth
difference equations a lot,
1165
01:11:43,820 --> 01:11:45,900
here it's easy.
1166
01:11:45,900 --> 01:11:49,480
In fact, you can see
it in the network.
1167
01:11:49,480 --> 01:11:51,500
Thinking about the difference
equation would be easy.
1168
01:11:51,500 --> 01:11:53,500
Thinking about the network
would be easy.
1169
01:11:53,500 --> 01:11:55,270
If we think about the
unit-sample response of this
1170
01:11:55,270 --> 01:12:00,330
thing started at rest, rest
means this is 0, this is 0,
1171
01:12:00,330 --> 01:12:03,270
and this is 0 initially.
1172
01:12:03,270 --> 01:12:08,060
Unit-sample response means
this becomes 1 at time 0.
1173
01:12:08,060 --> 01:12:13,000
At time 0 this is 1, this is
0, this is 0, this is 0.
1174
01:12:13,000 --> 01:12:27,070
So if I think about what's the
time response look like and I
1175
01:12:27,070 --> 01:12:31,820
did plus, this is
0, 1, 0, 0, 0.
1176
01:12:31,820 --> 01:12:33,610
The first answer is 0.
1177
01:12:33,610 --> 01:12:34,170
Clock ticks.
1178
01:12:34,170 --> 01:12:35,420
What happens?
1179
01:12:35,420 --> 01:12:38,450
1180
01:12:38,450 --> 01:12:40,394
This is 1.
1181
01:12:40,394 --> 01:12:43,630
This becomes 1.
1182
01:12:43,630 --> 01:12:44,570
This doesn't change.
1183
01:12:44,570 --> 01:12:45,510
That doesn't change.
1184
01:12:45,510 --> 01:12:47,620
This goes to 0.
1185
01:12:47,620 --> 01:12:48,640
0 comes around here.
1186
01:12:48,640 --> 01:12:50,270
That goes to 0.
1187
01:12:50,270 --> 01:12:52,640
So that's the answer
at time 1.
1188
01:12:52,640 --> 01:12:55,542
What happens at time 2?
1189
01:12:55,542 --> 01:12:57,070
Just keep working it.
1190
01:12:57,070 --> 01:13:01,850
The clock ticks, this goes to 1,
this goes to 0, these stay
1191
01:13:01,850 --> 01:13:05,330
0, this stays 0.
1192
01:13:05,330 --> 01:13:07,870
That's the answer at
time equals 2.
1193
01:13:07,870 --> 01:13:09,790
Now the clock ticks.
1194
01:13:09,790 --> 01:13:13,480
Now this comes over to here,
that means it comes back here.
1195
01:13:13,480 --> 01:13:14,450
This is still 0.
1196
01:13:14,450 --> 01:13:17,750
That comes to 1.
1197
01:13:17,750 --> 01:13:19,520
That's the answer for time 3.
1198
01:13:19,520 --> 01:13:20,940
Now the clock ticks.
1199
01:13:20,940 --> 01:13:22,010
And you can see the whole
thing would just
1200
01:13:22,010 --> 01:13:24,000
repeat itself now.
1201
01:13:24,000 --> 01:13:27,170
Is the response periodic?
1202
01:13:27,170 --> 01:13:28,070
Yes.
1203
01:13:28,070 --> 01:13:29,200
The response is periodic.
1204
01:13:29,200 --> 01:13:31,118
What's the period?
1205
01:13:31,118 --> 01:13:32,086
3.
1206
01:13:32,086 --> 01:13:34,506
And I can see [INAUDIBLE] if
the [INAUDIBLE] there it's
1207
01:13:34,506 --> 01:13:37,530
going to have be related to
the period over here.
1208
01:13:37,530 --> 01:13:39,290
These periods are not same.
1209
01:13:39,290 --> 01:13:42,300
This period is 3, this period is
3, this period is 1, if you
1210
01:13:42,300 --> 01:13:44,720
want to call that a period.
1211
01:13:44,720 --> 01:13:47,390
But they are related.
1212
01:13:47,390 --> 01:13:47,780
OK.
1213
01:13:47,780 --> 01:13:52,900
The point of this exercise is
to illustrate two things.
1214
01:13:52,900 --> 01:13:57,680
We inferred properties of first
order systems by looking
1215
01:13:57,680 --> 01:14:01,700
at a single pole which for a
real system could only behave
1216
01:14:01,700 --> 01:14:04,570
in one of four different ways.
1217
01:14:04,570 --> 01:14:06,810
Second order system introduced
new behaviors.
1218
01:14:06,810 --> 01:14:08,885
Now we can oscillate, which
we couldn't do before.
1219
01:14:08,885 --> 01:14:11,540
1220
01:14:11,540 --> 01:14:14,170
Having got to oscillation,
oscillation came about because
1221
01:14:14,170 --> 01:14:16,060
of complex numbers.
1222
01:14:16,060 --> 01:14:18,130
If you go to higher order
systems nothing
1223
01:14:18,130 --> 01:14:19,380
new happens in algebra.
1224
01:14:19,380 --> 01:14:22,360
1225
01:14:22,360 --> 01:14:25,490
There's no such thing as
meta-complex numbers, right?
1226
01:14:25,490 --> 01:14:28,200
Complex is as bad as it gets.
1227
01:14:28,200 --> 01:14:30,720
So you can have complex
numbers.
1228
01:14:30,720 --> 01:14:34,870
The higher order behaviors can
still have complex numbers but
1229
01:14:34,870 --> 01:14:38,360
you have to think when we ask
you, what's the property of a
1230
01:14:38,360 --> 01:14:39,700
higher order system.
1231
01:14:39,700 --> 01:14:42,620
You can think about it in terms
of the individual parts
1232
01:14:42,620 --> 01:14:45,320
but it requires some thinking.
1233
01:14:45,320 --> 01:14:45,740
OK.
1234
01:14:45,740 --> 01:14:46,710
Good luck tonight.
1235
01:14:46,710 --> 01:14:47,960
See you then.
1236
01:14:47,960 --> 01:14:53,503